Mathematical Sciences: Non-Locally Convex Spaces and Their Applications in Analysis

卡尔顿教授的研究项目继续他对在分析中遇到的相当高抽象水平的数学对象的研究。在这门学科最具体的层面上，人们关心的是级数求和、积分函数、解微分方程式等。一般而言，人们发现将正在处理的分析对象合并到所谓的Banach空间(在最有利的情况下)或其他类型的线性空间中是有利的。反过来，这些空间既可以单独研究，也可以按照一般理论的思路进行研究。本项目涉及与Banach空间相关但严格不属于该类别的空间。例如，有一种被称为扭曲和的结构，它在比Banach空间大得多的泛函分析对象中被最自然地看待。后者中的两个可以有一个甚至不是局部凸的扭曲和。卡尔顿将研究关于扭和定义的分解和扩张性质，以及这一概念与内插理论的联系。他的方法预计将对通常仅从Banach空间背景下处理的问题产生影响。

Professor Kalton's research project continues his investigation of mathematical objects encountered at a fairly high level of abstraction in analysis. At the most concrete level of this subject, one is concerned with summing series, integrating functions, solving differential equations, and the like. Moving up a notch in generality, it has been found to be advantageous to amalgamate the analytic objects one is dealing with into what are called Banach spaces (in the most favorable situations) or other sorts of linear spaces. These spaces in turn can be studied either individually, or along the lines of a general theory. The present project deals with spaces that are related to Banach spaces but do not fit strictly within that category. There is for instance a construction called the twisted sum that is most naturally viewed in a much larger category of functional analytic objects than that of Banach spaces. Two of the latter can have a twisted sum that is not even locally convex. Kalton will study decomposition and extension properties defined with respect to twisted sum, and the connections of this notion with interpolation theory. His methods are expected to have repercussions for problems usually approached just from within the Banach space context.